Investigation into Solvable Quintics

Abstract

This paper focuses on two families of quintics that pose different challenges for solving them. The first family is a famous group of quintics that are called Emma Lehmer's Quintics. These quintics are known to have the cyclic group of order 5 as their Galois group and one might hope that expressing the roots in terms of radicals would give simple expressions from which Emma Lehmer's polynomials could be recovered. However, we show that the expresions of the roots in terms of radicals is much more complicated than expected. We also consider the simpple equation f(x)=x^5+ax+p and show that, for a fixed nonzero integer p, the polynomial f is solvable by radicals for only finitely many integers a. David Dummit in Solving Solvable Quintics gives a powerful method that permits one to determine when a quintic is solvable and to solve for its roots. We will use Dummit's method to investigate both families of quintics.